Calculus I, Derivative
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Instantaneous rate of change
Suppose y is a quantity that depends on another quantity x. That is, y
can be viewed as a function y (x) of x.
If x changes from x1 to x2 then the change in x is
∆x = x2 − x1 .
The corresponding change in y is
∆y = y (x2 ) − y (x1 ).
The average rate of change of y with respect to x over the interval [x1 , x2 ]
is
∆y
y (x2 ) − y (x1 )
=
.
∆x
x2 − x1
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Instantaneous rate of change
Now, as we consider the average rate of change over smaller and smaller
intervals by letting ∆x approach 0,
the limit of these average rates of change is called instantaneous rate of
change of y with respect to x at x = x1
instantaneous rate of change
of y with respect to x
=
lim
∆x→0
∆y
∆x
Example
The cost (in dollars) of producing x units of a certain commodity is
C (x) = 4000 + 12x + 0.15x 2 .
(a) Find the average rate of change of C with respect to x when the
production is changed from x = 100 to x = 101.
(b) Find the instantaneous rate of change of C with respect to x when
x = 100. (This is called the marginal cost.)
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Definition of the derivative
Definition
The derivative of a function f (x) at a ∈ R, denoted f ′ (a) is the value of the
limit
f (x) − f (a)
f ′ (a) = lim
x→a
x −a
if the limit exists.
The above definition can be also rewritten as follows
f ′ (a) = lim
h→0
f (a + h) − f (a)
.
h
Notation: if y = f (x) then
f ′ (a) =
df
dy
d
(a) =
(a) =
f (a) = Df (a) = Dx f (a).
dx
dx
dx
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Geometric interpretation
Fact
f ′ (a) is the slope of the tangent line to the graph of f (x) at the point (a, f (a)).
The equation of the tangent line is
y − f (a) = f ′ (a) · (x − a).
Example
Find an equation of the tangent line to the graph of the function f (x) =
the point (3, 1).
1
x
at
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Obstacles to differentiability
f (x) is not differentiable at x = a if
the graph of f (x) has a “corner” at x = a
Example
Consider the function f (x) = |x|. It is not differentiable at x = 0.
In fact

if x > 0,
 1,
undefined if x = 0,
f ′ (x) =

−1,
if x < 0
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Obstacles to differentiability
f (x) is not differentiable at x = a if
f (x) is discontinuous at x = a
Example
The function
f (x) =
2x,
2x + 1,
if x 6 1,
if x > 1,
is not differentiable at x = 1.
Theorem
If f (x) is differentiable at x = a then it is continuous at x = a.
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Obstacles to differentiability
f (x) is not differentiable at x = a if
the tangent line to the graph of f (x) at x = a is vertical,
in other words
lim
h→0
f (a + h) − f (a)
= ±∞
h
Example
The function f (x) =
√
3
x is not differentiable at x = 0.
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Derivative as a function
If in the definition of the derivative the number a is not fixed, that is, it can be
viewed as a variable x then
f ′ (x) = lim
h→0
f (x + h) − f (a)
h
defines f ′ (x) as a function of x. Using this definition one can compute f ′ (x)
from the limit.
Example
f (x) =
1−x
−3
, f ′ (x) =
.
2+x
(2 + x)2
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Higher derivatives
One can take the derivative of the derivative and continue this process.
Example
f (x) = x 2 , f ′ (x) = 2x, f ′′ (x) = 2.
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Preliminary curve-sketching
Definition
f (x) is increasing on an interval if f (x1 ) < f (x2 ) for any points of the interval
x1 , x2 such that x1 < x2 .
f (x) is decreasing on an interval if f (x1 ) > f (x2 ) for any points of the interval
x1 , x2 such that x1 < x2 .
Fact
If f ′ (x) > 0 (f ′ (x) < 0) on an interval then f (x) is increasing (decreasing) on
that interval.
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Preliminary curve-sketching
Example
(a) If it is known that the graph of the derivative f ′ (x) of a function is as
shown below, what can we say about f ?
y
y = f '(x)
1
-1
0
1
x
-1
(b) If it is known that f (0) = 0, sketch a possible graph of f .
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Preliminary curve-sketching
Definition
The graph of a function f (x) is concave upward (downward) on an interval if it
is above (below) the tangent line at every point of that interval.
Fact
If f ′′ (x) > 0 (f ′′ (x) < 0) on an interval then the graph of f (x) is concave
upward (downward) on that interval.
Example
Sketch a possible graph of a function f that satisfies the following conditions:
f ′ (x) > 0 on (−∞, 1) and f ′ (x) < 0 on (1, ∞),
f ′′ (x) > 0 on (−∞, −2) and (2, ∞), f ′′ (x) < 0 on (−2, 2),
limx→−∞ f (x) = −2, limx→∞ f (x) = 0
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